According to this post here, there is a theorem by Ruzsa (2009) that for any group $G$ and $A,B\subset G$, $|A+B| \ge \min\{ p(G), |A|+|B|-1\}$ where $p(G)$ is the size of the smallest subgroup of $G$.
I cannot find this anywhere on the internet. I've read his notes from conferences since 2009 about sum sets and there is nothing about this. Could someone help me find the proof?
I am not an expert in the field but it seems like you are looking for the Cauchy-Davenport theorem generalization for an arbitrary group $G$. Here you can find the theorem for arbitrary finite groups.